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sav07_lecture_3_skeleton [2007/03/20 14:27] vkuncak |
sav07_lecture_3_skeleton [2007/03/20 14:35] vkuncak |
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| * efficient | * efficient | ||
| * create formulas that we can prove later | * create formulas that we can prove later | ||
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| ==== Formulas for basic statements ==== | ==== Formulas for basic statements ==== | ||
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| R(x=t) = (x=t & y=y_0 & error=error_0) | R(x=t) = (x=t & y=y_0 & error=error_0) | ||
| - | <b>Note</b>: all our statements will have the property that if error_0 = true, then error=true. That is, you can never recover from an error state. This is convenient: if we prove no errors at the end, then there were never errors in between. | + | **Note**: all our statements will have the property that if error_0 = true, then error=true. That is, you can never recover from an error state. This is convenient: if we prove no errors at the end, then there were never errors in between. |
| - | <b>Note</b>: the condition y=y_0 & error=error_0 is called <b>frame condition</b>. There are as many conjuncts as there are components of the state. This can be annoying to write, so let us use shorthand frame(x) for it. The shorthand frame(x) denotes a conjunction of v=v_0 for all v that are distinct from x (in this case y and error). We can have zero or more variables as arguments of frame, so frame() means that nothing changes. | + | **Note**: the condition y=y_0 & error=error_0 is called <b>frame condition</b>. There are as many conjuncts as there are components of the state. This can be annoying to write, so let us use shorthand frame(x) for it. The shorthand frame(x) denotes a conjunction of v=v_0 for all v that are distinct from x (in this case y and error). We can have zero or more variables as arguments of frame, so frame() means that nothing changes. |
| R(havoc x) = frame(x) | R(havoc x) = frame(x) | ||
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| R(assert F) = (F -> frame) | R(assert F) = (F -> frame) | ||
| - | Note: | + | **Note**: |
| x=t is same as havoc(x);assume(x=t) | x=t is same as havoc(x);assume(x=t) | ||
| assert false = crash (stops with error) | assert false = crash (stops with error) | ||
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| assume true = skip (does nothing) | assume true = skip (does nothing) | ||
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| + | ==== Composing formulas using relation composition ==== | ||
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| + | This is perhaps the most direct way of transforming programs to formulas. It creates formulas that are linear in the size of the program. | ||
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| + | Non-deterministic choice is union of relations, that is, disjunction of formulas: | ||
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| + | CR(c1 [] c2) = CR(c1) | CR(c2) | ||
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| + | In sequential composition we follow the rule for composition of relations. We want to get again formula with free variables x_0,y_0,x,y. So we need to do renaming. Let x_1,y_1,error_1 be fresh variables. | ||
| + | |||
| + | CR(c1 ; c2) = exists x_1,y_1,error_1. CR(c1)[x:=x_1,y:=y_1,error:=error_1] & CR(c2)[x:=x_1,y:=y_1,error:=error_1] | ||
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| ==== Papers ==== | ==== Papers ==== | ||